> I have two time scales, TAI and UT1, that tick at very slightly
> different rates. I want to make TAI the basis for civil time keeping
> but I need to make adjustments occasionally to keep it in step with
> UT1. How do I do it?
>
> The answer provided by CCIR was to represent TAI in a variable-radix
> notation that matches (or appears to match), to within 0.9s, that of
> UT1 expressed in the usual calendar/clock format. This is done by
> varying the radix of the seconds field in a pseudo-sexagesimal clock
> format from 60 to 61 (or in principle 59) on occasions announced 6
> months in advance.
>
> So if asked for a definition I would say that "UTC (post 1972) is a
> representation of TAI such that ... (you know the rest)".
>
> The point is that UTC is simply a representation of TAI. "Writing UTC
> as a real" reveals it to be TAI.

I believe I'm now grasping what you mean: the rate of UTC is the same
as the rate of TAI (since 1972), that is, the derivative
d( UTC )/d( TAI ) = 1. Hence, when I integrate the "ticks" of UTC
I must get TAI, up to an integration constant. This is correct.
The integral of d( UTC ) is TAI (up to an integration constant),
but this integral is UTC only where UTC is a continuous function
of TAI.

Astronomers who "write UTC as a real" (eg, in JD or MJD notation)
want an approximation of UT1 to point their telescopes, they do
_not_ want TAI. They use UTC as a timescale whose values are
close to UT1, but whose rate nevertheless is d( UTC ) = d( TAI )
and not d( UT1 ). Such a function cannot be continuous (and it
cannot be differentiable everywhere). At the latest discontinuity
of UTC, it jumped from a little bit after UT1 to a little bit before
UT1.